RICCI CURVATURE FOR METRIC MEASURE SPACES VIA OPTIMAL TRANSPORT

Description

Niveau: Supérieur, Doctorat, Bac+8
RICCI CURVATURE FOR METRIC-MEASURE SPACES VIA OPTIMAL TRANSPORT JOHN LOTT AND CEDRIC VILLANI Abstract. We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ? [1,∞), or having ∞-Ricci curvature bounded below by K, for K ? R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov–Hausdorff limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X, d) in which the distance between two points equals the infimum of the lengths of curves joining the points.

• original metric


• locally compact


• length spaces


• ricci curvature


• riemannian manifold


• transport


• bounded below


• optimal transport


• compact measured


• dimensional riemannian

Sujets

• Locally compact space
• Ricci curvature
• Riemannian manifold
• Transport
• Transportation theory

Informations

• Publié par: mijec
• Nombre de lectures: 73
• Langue: English

Extrait

RICCICURVATUREFORMETRIC-MEASURESPACESVIAOPTIMAL
TRANSPORT

JOHNLOTTANDCE´DRICVILLANI

Abstract.
Wedefineanotionofameasuredlengthspace
X
havingnonnegative
N
-Ricci
curvature,for
N
∈
[1
,
∞
),orhaving
∞
-Riccicurvatureboundedbelowby
K
,for
K
∈
R
.
Thedefinitionsareintermsofthedisplacementconvexityofcertainfunctionsonthe
associatedWassersteinmetricspace
P
2
(
X
)ofprobabilitymeasures.Weshowthatthese
propertiesarepreservedundermeasuredGromov–Hausdorfflimits.Wegivegeometric
andanalyticconsequences.

Thispaperhasdualgoals.Onegoalistoextendresultsaboutoptimaltransportfromthe
settingofsmoothRiemannianmanifoldstothesettingoflengthspaces.Asecondgoalisto
useoptimaltransporttogiveanotionforameasuredlengthspacetohaveRiccicurvature
boundedbelow.Wereferto[11]and[44]forbackgroundmaterialonlengthspacesand
optimaltransport,respectively.Furtherbibliographicnotesonoptimaltransportarein
AppendixF.Inthepresentintroductionwemotivatethequestionsthatweaddressand
westatethemainresults.
Tostartonthegeometricside,therearevariousreasonstotrytoextendnotionsof
curvaturefromsmoothRiemannianmanifoldstomoregeneralspaces.Afairlygeneral
settingisthatof
lengthspaces
,meaningmetricspaces(
X,d
)inwhichthedistancebetween
twopointsequalstheinfimumofthelengthsofcurvesjoiningthepoints.Intherestof
thisintroductionweassumethat
X
isacompactlengthspace.Alexandrovgaveagood
notionofalengthspacehaving
“curvatureboundedbelowby
K
”
,with
K
arealnumber,
intermsofthegeodesictrianglesin
X
.InthecaseofaRiemannianmanifold
M
withthe
inducedlengthstructure,onerecoverstheRiemanniannotionofhavingsectionalcurvature
boundedbelowby
K
.LengthspaceswithAlexandrovcurvatureboundedbelowby
K
behavenicelywithrespecttotheGromov–Hausdorfftopologyoncompactmetricspaces
(moduloisometries);theyformaclosedsubset.
InviewofAlexandrov’swork,itisnaturaltoaskwhethertherearemetricspaceversions
ofothertypesofRiemanniancurvature,suchasRiccicurvature.Thisquestiontakes
substancefrom
Gromov’sprecompactnesstheorem
forRiemannianmanifoldswithRicci
curvatureboundedbelowby
K
,dimensionboundedaboveby
N
anddiameterbounded
aboveby
D
[23,Theorem5.3].Theprecompactnessindicatesthattherecouldbeanotion
ofalengthspacehaving“Riccicurvatureboundedbelowby
K
”,specialcasesofwhich
wouldbeGromov–HausdorfflimitsofmanifoldswithlowerRiccicurvaturebounds.
Date
:June23,2006.
TheresearchofthefirstauthorwassupportedbyNSFgrantDMS-0306242andtheClayMathematics
Institute.
1

2JOHNLOTTANDCE´DRICVILLANI

Gromov–HausdorfflimitsofmanifoldswithRiccicurvatureboundedbelowhavebeen
studiedbyvariousauthors,notablyCheegerandColding[15,16,17,18].Onefeatureof
theirwork,alongwiththeearlierworkofFukaya[21],isthatitturnsouttobeusefulto
addanauxiliaryBorelprobabilitymeasure
ν
andconsider
metric-measurespaces
(
X,d,ν
).
(AcompactRiemannianmanifold
M
hasacanonicalmeasure
ν
givenbythenormalized
Riemanniandensity
vdovl(ol
M
M
)
.)ThereisameasuredGromov–Hausdorfftopologyonsuch
triples(
X,d,ν
)(moduloisometries)andoneagainhasprecompactnessforRiemannian
manifoldswithRiccicurvatureboundedbelowby
K
,dimensionboundedaboveby
N
and
diameterboundedaboveby
D
.Hencethequestioniswhetherthereisagoodnotionofa
measuredlengthspace(
X,d,ν
)having
“Riccicurvatureboundedbelowby
K
”
.Whatever
definitiononetakes,onewouldlikethesetofsuchtriplestobeclosedinthemeasured
Gromov–Hausdorfftopology.Onewouldalsoliketoderivesomenontrivialconsequences
fromthedefinition,andofcourseinthecaseofRiemannianmanifoldsonewouldliketo
recoverclassicalnotions.Wereferto[16,Appendix2]forfurtherdiscussionoftheproblem
ofgivinga“synthetic”treatmentofRiccicurvature.
Ourapproachisintermsofametricspace(
P
(
X
)
,W
2
)thatiscanonicallyassociatedto
theoriginalmetricspace(
X,d
).Here
P
(
X
)isthespaceofBorelprobabilitymeasureson
X
and
W
2
istheso-called
Wassersteindistance
oforder2.ThesquareoftheWasserstein
distance
W
2
(

0
,
1
)between

0
,
1
∈
P
(
X
)isdefinedtobetheinfimalcosttotransport
thetotalmassfromthemeasure

0
tothemeasure

1
,wherethecosttotransportaunit
ofmassbetweenpoints
x
0
,x
1
∈
X
istakentobe
d
(
x
0
,x
1
)
2
.Atransportationscheme
withinfimalcostiscalledan
optimaltransport
.Thetopologyon
P
(
X
)comingfromthe
metric
W
2
turnsouttobetheweak-
∗
topology.Wewillwrite
P
2
(
X
)forthemetricspace
(
P
(
X
)
,W
2
),whichwecallthe
Wassersteinspace
.If(
X,d
)isalengthspacethen
P
2
(
X
)
turnsouttoalsobealengthspace.Itsgeodesicswillbecalled
Wassersteingeodesics
.If
M
isaRiemannianmanifoldthenwewrite
P
2
ac
(
M
)fortheelementsof
P
2
(
M
)thatare
absolutelycontinuouswithrespecttotheRiemanniandensity.
Inthepastfifteenyears,optimaltransportofmeasureshasbeenextensivelystudiedin
thecase
X
=
R
n
,withmotivationcomingfromthestudyofcertainpartialdifferential
equations.Anotionwhichhasprovedusefulisthatof
displacementconvexity
,i.e.convex-
ityalongWassersteingeodesics,whichwasintroducedbyMcCanninordertoshowthe
existenceanduniquenessofminimizersforcertainrelevantfunctionson
P
2
ac
(
R
n
)[31].
Inthepastfewyears,someregularityresultsforoptimaltransporton
R
n
havebeen
extendedtoRiemannianmanifolds[19,32].Thismadeitpossibletostudydisplacement
convexityinaRiemanniansetting.OttoandVillani[36]carriedoutHessiancomputations
forcertainfunctionson
P
2
(
M
)usingaformalinfinite-dimensionalRiemannianstructure
on
P
2
(
M
)definedbyOtto[35].Theseformalcomputationsindicatedarelationshipbe-
tweentheHessianofan“entropy”functionon
P
2
(
M
)andtheRiccicurvatureof
M
.Later,
arigorousdisplacementconvexityresultforaclassoffunctionson
P
2
ac
(
M
),when
M
has
nonnegativeRiccicurvature,wasprovenbyCordero-Erausquin,McCannandSchmucken-
schla¨ger[19].ThisworkwasextendedbyvonRenesseandSturm[40].

RICCICURVATUREVIAOPTIMALTRANSPORT3
AgaininthecaseofRiemannianmanifolds,afurthercircleofideasrelatesdisplacement
convexitytologSobolevinequalities,Poincare´inequalities,Talagrandinequalitiesand
concentrationofmeasure[8,9,27,36].
Inthispaperweuseoptimaltransportanddisplacementconvexityinorderto
define
a
notionofameasuredlengthspace(
X,d,ν
)havingRiccicurvatureboundedbelow.If
N
is
afiniteparameter(playingtheroleofadimension)thenwewilldefineanotionof(
X,d,ν
)
havingnonnegative
N
-Riccicurvature.Wewillalsodefineanotionof(
X,d,ν
)having
∞
-Riccicurvatureboundedbelowby
K
∈
R
.(Theneedtoinputthepossibly-infinite
parameter
N
canbeseenfromtheBishop–Gromovinequalityforcomplete
n
-dimensional
RiemannianmanifoldswithnonnegativeRiccicurvature.Itstatesthat
r
−
n
vol(
B
r
(
m
))is
nonincreasingin
r
,where
B
r
(
m
)isthe
r
-ballcenteredat
m
[23,Lemma5.3.bis].When
wegofrommanifoldstolengthspacesthereisno
apriori
valuefortheparameter
n
.This
indicatestheneedtospecifyadimensionparameterinthedefinitionofRiccicurvature
bounds.)
Wenowgivethemainresultsofthepaper,sometimesinasimplifiedform.Forconsis-
tency,weassumeinthebodyofthepaperthattherelevantlengthspace
X
iscompact.
Thenecessarymodificationstodealwithcompletepointedlocallycompactlengthspaces
aregiveninAppendixE.
Let
U
:[0
,
∞
)
→
R
beacontinuousconvexfunctionwith
U
(0)=0.Givenareference
probabilitymeasure
ν
∈
P
(
X
),definethefunction
U
ν
:
P
2
(
X
)
→
R
∪{∞}
by
Z(0.1)
U
ν
(

)=
U
(
ρ
(
x
))
dν
(
x
)+
U
′
(
∞
)

s
(
X
)
,
Xerehw(0.2)

=
ρν
+

s
istheLebesguedecompositionof

withrespectto
ν
intoanabsolutelycontinuouspart
ρν
andasingularpart

s
,and
(0.3)
U
′
(
∞
)=lim
U
(